Interdependent spatially embedded networks: Dynamics at percolation threshold

Spatially embedded systems of interdependent networks with full dependency (q=1) have been found to have a first-order percolation transition if the dependency link length (the maximum distance in lattice units between a node in one network and the node that it depends on in another network) is longer than a certain critical length r-capprox 8. We find here that a similar result is valid for any finite value of q with a larger r-c as q decreases. We also provide a theoretical approach which correctly predicts the relationship between r-c and q. We also examine the dynamics at the percolation threshold p-c for varying r and q and find that there are three different mechanisms of failure for every q depending on r. Below r-c the system undergoes a continuous transition similar to standard percolation. Above r-c there are two distinct first-order transitions for finite or infinite r, respectively. The transition for finite r is characterized by spreading of node failures through the system while the infinite r corresponds to a non-spatial cascading failure similar to the case of random networks. These results extend previous results on spatially embedded interdependent networks to the more realistic cases of partial dependency and shed new light on the specific dynamics of cascading failures in such systems.